Kelvin–Helmholtz instability of Couette flow between vertical walls with a free surface

Abstract

A novel type of Kelvin–Helmholtz instability model is developed from hydrodynamic theory. The classical Kelvin–Helmholtz instability involves a horizontal interface between two fluids with different parallel, uniform, horizontal velocities. If the upper fluid is a gas with a much smaller density than the lower fluid which is a liquid, then the phase velocity of the critical disturbance equals the liquid’s velocity, so that the liquid sees a standing interfacial wave. The inertial force driving the interfacial instability involves only the gas, no matter how small its density is. In a much more realistic flow model, the liquid velocity at the free surface is not uniform, but varies across the free surface. The disturbance phase velocity can only equal the liquid velocity at one point, while liquid on either side of this point moves faster or slower than the wave. The inertial forces in the liquid then dominate and the gas plays a negligible role. The concept is developed from a Couette flow hydrodynamic model where the fluid flows between two parallel vertical walls with a free surface. The importance of a nonuniform liquid velocity is demonstrated. This modified theory will be applied in future work to study the ejection instability at the interface of the liquid metal and inert cover gas in sliding electrical contacts.